If a=2^1/3 - 2^-1/3 , then the value of 2a^3+6a is
Keerthana632:
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x = 2^(1/3) - 2^(-1/3)
= 2^(1/3) - 1/2^(1/3)
= (2^(2/3) -1)/2^(1/3)
therefore, 2x^3 + 6x
= 2 [2x^(2/3) -1]^3 + 6(2^(2/3) -1)/2^(1/3)
= 2^2 -1 -3*2^(2/3)(2^(2/3) -1) + 6*2^(2/3)(2^(2/3) -1)
= 3 - 3 * 2 ^(4/3) + 3*2^(2/3) + 3*2^(2/3)(2^(2/3)-1)
= 3 - 3*2*2^(1/3) + 3*2^(2/3) + 3*2*2^(1/3) - 3*2^(2/3)
= 3
= 2^(1/3) - 1/2^(1/3)
= (2^(2/3) -1)/2^(1/3)
therefore, 2x^3 + 6x
= 2 [2x^(2/3) -1]^3 + 6(2^(2/3) -1)/2^(1/3)
= 2^2 -1 -3*2^(2/3)(2^(2/3) -1) + 6*2^(2/3)(2^(2/3) -1)
= 3 - 3 * 2 ^(4/3) + 3*2^(2/3) + 3*2^(2/3)(2^(2/3)-1)
= 3 - 3*2*2^(1/3) + 3*2^(2/3) + 3*2*2^(1/3) - 3*2^(2/3)
= 3
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